On the principle that it is easier to ask questions than to answer them: could you please edit your question and give a bit more information on what you've already tried to do towards answering this problem? For instance, why do you think it might have simple spectrum?
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Yemon ChoiOct 8 '10 at 2:08

3 Answers
3

Yes, I believe that it will have simple spectrum for d >= 3 and it feels like something that should have been proved, though I can't actually find it.

There is a loose association between automorphisms of a graph and multiple eigenvalues, and as most regular graphs have trivial automorphism group we lose this source of multiple eigenvalues. There are no other (frequently occurring) "reasons" for a graph to have multiple eigenvalues and so in general they won't be there.

ADDED:

Here are some exact numbers for connected (pairwise nonisomorphic) cubic graph: disconnected graphs vanish numerically and so we can ignore them.

I guessed no but Gordon has changed my mind for degree greater than 2.

If it has degree 2 it is a union of cycles. Eigenvalues are $2\cos(\frac{2 \pi j}{k})$ for various $ k$. In particular 2 has multiplicity the number of components.

For random regular graphs of degree more than 2 I'd wildly guess that the eigenvalue behavior is like that of a large random symmetric matrix. This has been well studied, but not by me, all I know is the phrase "Gaussian Orthogonal Ensemble".

Some experiments with degree 3 graphs suggest that with 30 vertices one component is highly likely and there is a repeated eigenvalue ( most often 0) about 2% of the time. At 60 vertices it is more like 0.2%.

Think you're right: arxiv.org/abs/hep-th/0310002 , which would make it follow Wigner's Semi Circle law. As a heuristic, you can think of Erdos-Renyi random graphs as essentially being regular in their mean degree, so it's not so surprising that actual regular graphs have the same property...
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dorkusmonkeyOct 8 '10 at 15:14

In 1986, Noga Alon conjectured
in a Combinatorica article that,
for any degree $d \ge 3$ and for any $\epsilon > 0$,
most $d$-regular graphs on $n$ vertices have all their eigenvalues except $\lambda_1=d$
bounded above by $2 \sqrt{d−1} + \epsilon$.
In 2003,
Joel Friedman established this conjecture:
"A proof of Alon's second eigenvalue conjecture,"
2003.
Some further developments on the distribution of the eigenvalues are reported
in a paper by Miller, Novikoff, and Anthony Sabelli,
"The Distribution of the Largest Nontrivial Eigenvalues in Families of Random Regular Graphs"
in Experimental Mathematics, 2008. Perhaps some of these references will help.